Potential operators in modified Morrey spaces defined on Carleson curves

Abstract

In this paper we study the potential operator $I_{\Gamma }^{\alpha }$, $0<1$ in the modified Morrey space ${\tilde{L}}_{p,\lambda }\left(\Gamma \right)$ and the spaces $BMO\left(\Gamma \right)$ defined on Carleson curves $\Gamma$. We prove that for $1<p<(1-\lambda )∕\alpha$ the potential operator $I_{\Gamma }^{\alpha }$ is bounded from the modified Morrey space ${\tilde{L}}_{p,\lambda }\left(\Gamma \right)$ to ${\tilde{L}}_{q,\lambda }\left(\Gamma \right)$ if and in the case of infinite curve only if $\alpha \le 1∕p-1∕q\le \alpha ∕(1-\lambda )$, and from the spaces ${\tilde{L}}_{1,\lambda }\left(\Gamma \right)$ to $W{\tilde{L}}_{q,\lambda }\left(\Gamma \right)$ if and in the case of infinite curve only if $\alpha \le 1-\frac{1}{q}\le \frac{\alpha }{1-\lambda }$. Furthermore, for the limiting case $(1-\lambda )∕\alpha \le p\le 1∕\alpha$ we show that if $\Gamma$ is an infinite Carleson curve, then the modified potential operator ${\tilde{I}}_{\Gamma }^{\alpha }$ is bounded from ${\tilde{L}}_{p,\lambda }\left(\Gamma \right)$ to $BMO\left(\Gamma \right)$, and if $\Gamma$ is a finite Carleson curve, then the operator $I_{\Gamma }^{\alpha }$ is bounded from ${\tilde{L}}_{p,\lambda }\left(\Gamma \right)$ to $BMO\left(\Gamma \right)$.